Rigid structures in the universal enveloping traffic space

Abstract

For any tracial non-commutative probability space $(\mathcal{A},\phi )$, Cébron, Dahlqvist, and Male showed that one can always construct an enveloping traffic space $(\mathcal{G}(\mathcal{A}),{\tau }_{\phi })$ that extends the trace. This construction provides a universal object that allows one to appeal to the traffic probability framework in generic situations, prioritizing an understanding of its structure. In this article, we prove that $(\mathcal{G}(\mathcal{A}),{\tau }_{\phi })$ admits a canonical free product decomposition $\mathcal{A}\ast {\mathcal{A}}^{⊺}\ast \Theta (\mathcal{G}(\mathcal{A}))$. In particular, ${\mathcal{A}}^{⊺}$ is an anti-isomorphic copy of $\mathcal{A}$, and $\Theta (\mathcal{G}(\mathcal{A}))$ is, up to degeneracy, a commutative algebra generated by Gaussian random variables with a covariance structure diagonalized by the graph operations. If $(\mathcal{A},\phi )$ itself is a free product, then we describe how this additional structure lifts into $(\mathcal{G}(\mathcal{A}),{\tau }_{\phi })$. Here, we find a connection between free independence and classical independence opposite the usual direction. Up to degeneracy, we further show that $(\mathcal{G}(\mathcal{A}),{\tau }_{\phi })$ is spanned by tree-like graph operations. Finally, we apply our results to the study of large (possibly dependent) random matrices. Our analysis relies on the combinatorics of cactus graphs and the resulting cactus-cumulant correspondence.

54 pages, 16 figures

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